# An infinite geometric series has a sum of 20, where all the terms are positive. The sum of the first and second terms are 12.8. What is the first term?

Mar 30, 2018

$a = 8$

#### Explanation:

Recall the sum of the terms in an infinite geometric series is $20$.

${S}_{n} = \frac{a}{1 - r}$

We know the sum of the series, so:

$20 = \frac{a}{1 - r}$

$a = 20 \left(1 - r\right)$

The first term is $a$. The second term is $a r$. Therefore, ${s}_{2} = a + a r = a \left(1 + r\right)$. We know the sum of the first two terms is $12.8 = \frac{64}{5}$. Therefore our second equation becomes $a \left(1 + r\right) = \frac{64}{5}$.

Substituting the first equation into the second we get:

$\left(20 \left(1 - r\right)\right) \left(1 + r\right) = \frac{64}{5}$

$\left(20 - 20 r\right) \left(1 + r\right) = \frac{64}{5}$

$20 - 20 r + 20 r - 20 {r}^{2} = \frac{64}{5}$

$100 - 100 {r}^{2} = 64$

$36 = 100 {r}^{2}$

$0.36 = {r}^{2}$

$r = \pm 0.6$

Since all the terms are positive, $r = + 0.6$ (if the common ratio was negative, every two terms would be negative).

Since $a = 20 \left(1 - r\right)$, $a = 20 \left(0.4\right) = 8$

Hopefully this helps!